User:H2g2bob/sandbox

This is a sandbox for Table of mathematical symbols

The following table lists many specialized symbols commonly used in mathematics.

Set theory

 * rowspan=3 bgcolor=#d0f0d0 align=center| $$\pm$$ $$\mp$$
 * plus-minus
 * rowspan=3|6 $$\pm$$ 3 means both 6 + 3 and 6 - 3. 6 $$\pm$$ 3 $$\mp$$ 5 means both 6 + 3 - 5 and 6 - 3 + 5.
 * rowspan=3|6 $$\pm$$ 3 = 9 or 3 6 $$\pm$$ 3 $$\mp$$ 5 = 4 or 8
 * align=center|plus-minus; plus-or-minus minus-plus; minus-or-plus
 * align=right|arithmetic
 * align=center|plus-minus; plus-or-minus minus-plus; minus-or-plus
 * align=right|arithmetic
 * align=right|arithmetic


 * rowspan=6 bgcolor=#d0f0d0 align=center| √
 * square root
 * rowspan=3|√x means the positive number whose square is x.
 * rowspan=3|√4 = 2
 * align=center|the principal square root of; square root
 * align=right|real numbers
 * align=center|the principal square root of; square root
 * align=right|real numbers
 * align=right|real numbers


 * complex square root
 * rowspan=3| if z = r exp(i&phi;) is represented in polar coordinates with -π < &phi; ≤ π, then √z = √r exp(i &phi;/2).
 * rowspan=3|√(-1) = i
 * align=center|the complex square root of … square root
 * align=right|complex numbers
 * align=center|the complex square root of … square root
 * align=right|complex numbers
 * align=right|complex numbers


 * rowspan=6 bgcolor=#d0f0d0 align=center| |…|
 * absolute value
 * rowspan=3| |x| means the distance along the real line (or across the complex plane) between x and zero.
 * rowspan=3| |3| = 3 |–5| = |5|  | i | = 1  | 3 + 4i | = 5
 * align=center|absolute value of
 * align=right|numbers
 * align=center|absolute value of
 * align=right|numbers
 * align=right|numbers


 * Euclidean distance
 * rowspan=3| |x – y| means the Euclidean distance between x and y.
 * rowspan=3| For x = (1,1), and y = (4,5), |x – y| = √([1–4]2 + [1–5]2) = 5
 * align=center|Euclidean distance between; Euclidean norm of
 * align=right|Geometry
 * align=center|Euclidean distance between; Euclidean norm of
 * align=right|Geometry
 * align=right|Geometry


 * rowspan=3 bgcolor=#d0f0d0 align=center| |
 * divides
 * rowspan=3| A single vertical bar is used to denote divisibility. a|b means a divides b.
 * rowspan=3| Since 15 = 3&times;5, it is true that 3|15 and 5|15.
 * align=center|divides
 * align=right|Number Theory
 * align=center|divides
 * align=right|Number Theory
 * align=right|Number Theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| !
 * factorial
 * rowspan=3|n ! is the product 1 &times; 2&times; ... &times; n.
 * rowspan=3|4! = 1 &times; 2 &times; 3 &times; 4 = 24
 * align=center|factorial
 * align=right|combinatorics
 * align=center|factorial
 * align=right|combinatorics
 * align=right|combinatorics


 * rowspan=3 bgcolor=#d0f0d0 align=center| T
 * transpose
 * rowspan=3| Swap rows for columns
 * rowspan=3| $$A_{ij} = (A^T)_{ji}$$
 * align=center|transpose
 * align=right|matrix operations
 * align=center|transpose
 * align=right|matrix operations
 * align=right|matrix operations


 * rowspan=3 bgcolor=#d0f0d0 align=center| ~
 * probability distribution
 * rowspan=3| X ~ D, means the random variable X has the probability distribution D.
 * rowspan=3|''X ~ N(0,1), the standard normal distribution
 * align=center|has distribution
 * align=right|statistics
 * align=center|has distribution
 * align=right|statistics
 * align=right|statistics


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⇒ →  ⊃
 * material implication
 * rowspan=3|A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B. → may mean the same as ⇒, or it may have the meaning for functions given below.  ⊃ may mean the same as ⇒, or it may have the meaning for superset given below.
 * rowspan=3|x = 2 ⇒  x2 = 4 is true, but x2 = 4   ⇒  x = 2 is in general false (since x could be &minus;2).
 * align=center|implies; if … then
 * align=right|propositional logic
 * align=center|implies; if … then
 * align=right|propositional logic
 * align=right|propositional logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⇔ ↔
 * material equivalence
 * rowspan=3|A ⇔ B means A is true if B is true and A is false if B is false.
 * rowspan=3|x + 5 = y +2 ⇔  x + 3 = y
 * align=center|if and only if; iff
 * align=right|propositional logic
 * align=center|if and only if; iff
 * align=right|propositional logic
 * align=right|propositional logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| ¬ ˜
 * logical negation
 * rowspan=3|The statement ¬A is true if and only if A is false. A slash placed through another operator is the same as "¬" placed in front.  (The symbol ~ has many other uses, so ¬  or the slash notation is preferred.)
 * rowspan=3|¬(¬A) ⇔ A x ≠ y ⇔  ¬(x =  y)
 * align=center|not
 * align=right|propositional logic
 * align=center|not
 * align=right|propositional logic
 * align=right|propositional logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| $$\wedge$$
 * logical conjunction or meet in a lattice
 * rowspan=3|The statement A $$\wedge$$ B is true if A and B are both true; else it is false. For functions A(x) and B(x), A(x) $$\wedge$$ B(x) is used to mean min(A(x), B(x)).
 * rowspan=3|n < 4 ∧  n >2  ⇔  n = 3 when n is a natural number.
 * align=center|and; min
 * align=right|propositional logic, lattice theory
 * align=center|and; min
 * align=right|propositional logic, lattice theory
 * align=right|propositional logic, lattice theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∨
 * logical disjunction or join in a lattice
 * rowspan=3|The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. For functions A(x) and B(x), A(x) ∨ B(x) is used to mean max(A(x), B(x)).
 * rowspan=3|n ≥ 4 ∨  n ≤ 2  ⇔ n ≠ 3 when n is a natural number.
 * align=center|or; max
 * align=right|propositional logic, lattice theory
 * align=center|or; max
 * align=right|propositional logic, lattice theory
 * align=right|propositional logic, lattice theory

⊕  ⊻ ||exclusive or
 * rowspan=6 bgcolor=#d0f0d0 align=center|
 * rowspan=6 bgcolor=#d0f0d0 align=center|
 * rowspan=3| The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same.
 * rowspan=3| (¬A) ⊕ A is always true, A ⊕ A is always false.
 * align=center|xor
 * align=right|propositional logic, Boolean algebra
 * direct sum
 * rowspan=3|The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, ⊻ is only for logic).
 * rowspan=3|Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = &empty;)
 * align=center|direct sum of
 * align=right|Abstract algebra
 * rowspan=3|Most commonly, for vector spaces U, V, and W, the following consequence is used: U = V ⊕ W ⇔ (U = V + W) ∧ (V ∩ W = &empty;)
 * align=center|direct sum of
 * align=right|Abstract algebra
 * align=right|Abstract algebra
 * align=right|Abstract algebra


 * rowspan=3 bgcolor=#d0f0d0 align=center| &forall;
 * universal quantification
 * rowspan=3|&forall; x: P(x) means P(x) is true for all x.
 * rowspan=3|&forall; n ∈ ℕ: n2 ≥ n.
 * align=center|for all; for any; for each
 * align=right|predicate logic
 * align=center|for all; for any; for each
 * align=right|predicate logic
 * align=right|predicate logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| &exist;
 * existential quantification
 * rowspan=3|&exist; x: P(x) means there is at least one x such that P(x) is true.
 * rowspan=3|&exist; n ∈ ℕ: n is even.
 * align=center|there exists
 * align=right|predicate logic
 * align=center|there exists
 * align=right|predicate logic
 * align=right|predicate logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| &exist;!
 * uniqueness quantification
 * rowspan=3|&exist;! x: P(x) means there is exactly one x such that P(x) is true.
 * rowspan=3|&exist;! n ∈ ℕ: n + 5 = 2n.
 * align=center|there exists exactly one
 * align=right|predicate logic
 * align=center|there exists exactly one
 * align=right|predicate logic
 * align=right|predicate logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| := ≡  :⇔
 * definition
 * rowspan=3|x := y or x ≡ y means x is defined to be another name for y (Some writers use ≡ to mean congruence).  P :⇔ Q means P is defined to be logically equivalent to Q.
 * rowspan=3|cosh x := (1/2)(exp x + exp (&minus;x))  A xor B :⇔ (A ∨ B) ∧ ¬(A ∧ B)
 * align=center|is defined as
 * align=right|everywhere
 * align=center|is defined as
 * align=right|everywhere
 * align=right|everywhere


 * rowspan=3 bgcolor=#d0f0d0 align=center| &cong;
 * congruence
 * rowspan=3|△ABC &cong; △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF.
 * rowspan=3|
 * align=center|is congruent to
 * align=right|geometry
 * align=center|is congruent to
 * align=right|geometry
 * align=right|geometry


 * rowspan=3 bgcolor=#d0f0d0 align=center| {, }
 * set brackets
 * rowspan=3|{a,b,c} means the set consisting of a, b, and c.
 * rowspan=3|ℕ = { 1, 2, 3, …}
 * align=center|the set of …
 * align=right|set theory
 * align=center|the set of …
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| { : } { | }
 * set builder notation
 * rowspan=3|{x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}.
 * rowspan=3|{n ∈ ℕ : n2 < 20} = { 1, 2, 3, 4}
 * align=center|the set of … such that
 * align=right|set theory
 * align=center|the set of … such that
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| &empty; { } ||empty set
 * rowspan=3|&empty; means the set with no elements. { } means the same.
 * rowspan=3|{n ∈ ℕ : 1 < n2 < 4} = &empty;
 * align=center| the empty set
 * align=right|set theory
 * align=center| the empty set
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∈
 * set membership
 * rowspan=3|a ∈ S means a is an element of the set S; a S means a is not an element of S.
 * rowspan=3|(1/2)&minus;1 ∈ ℕ 2&minus;1  ℕ
 * align=center|is an element of; is not an element of
 * align=right|everywhere, set theory
 * align=center|is an element of; is not an element of
 * align=right|everywhere, set theory
 * align=right|everywhere, set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⊆ ⊂
 * subset
 * rowspan=3|(subset) A ⊆ B means every element of A is also element of B. (proper subset) A ⊂ B means A ⊆ B but A ≠ B.  (Some writers use the symbol ⊂ as if it were the same as ⊆.)
 * rowspan=3|(A ∩ B) ⊆ A ℕ ⊂ ℚ  ℚ ⊂ ℝ
 * align=center|is a subset of
 * align=right|set theory
 * align=center|is a subset of
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⊇ ⊃
 * superset
 * rowspan=3|A ⊇ B means every element of B is also element of A. A ⊃ B means A ⊇ B but A ≠ B.  (Some writers use the symbol ⊃ as if it were the same as ⊇.)
 * rowspan=3|(A ∪ B) ⊇ B ℝ ⊃ ℚ
 * align=center|is a superset of
 * align=right|set theory
 * align=center|is a superset of
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∪
 * set-theoretic union
 * rowspan=3|(exclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, but not both. "A or B, but not both." (inclusive) A ∪ B means the set that contains all the elements from A, or all the elements from B, or all the elements from both A and B. "A or B or both".
 * rowspan=3|A ⊆ B ⇔  (A ∪ B) = B (inclusive)
 * align=center|the union of … and union
 * align=right|set theory
 * align=center|the union of … and union
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∩
 * set-theoretic intersection
 * rowspan=3|A ∩ B means the set that contains all those elements that A and B have in common.
 * rowspan=3|{x ∈ ℝ : x2 = 1} ∩ ℕ = {1}
 * align=center|intersected with; intersect
 * align=right|set theory
 * align=center|intersected with; intersect
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| $$\Delta$$
 * symmetric difference
 * rowspan=3|$$ A\Delta B$$ means the set of elements in exactly one of A or B.
 * rowspan=3|{1,5,6,8} $$\Delta$$ {2,5,8} = {1,2,6}
 * align=center|symmetric difference
 * align=right|set theory
 * align=center|symmetric difference
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∖
 * set-theoretic complement
 * rowspan=3|A ∖ B means the set that contains all those elements of A that are not in B.
 * rowspan=3|{1,2,3,4} ∖ {3,4,5,6} = {1,2}
 * align=center|minus; without
 * align=right|set theory
 * align=center|minus; without
 * align=right|set theory
 * align=right|set theory


 * rowspan=6 bgcolor=#d0f0d0 align=center|
 * function application
 * rowspan=3|f(x) means the value of the function f at the element x.
 * rowspan=3|If f(x) := x2, then f(3) = 32 = 9.
 * align=center|of
 * align=right|set theory
 * align=center|of
 * align=right|set theory
 * align=right|set theory


 * precedence grouping
 * rowspan=3|Perform the operations inside the parentheses first.
 * rowspan=3|(8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4.
 * align=center|parentheses
 * align=right|everywhere
 * align=center|parentheses
 * align=right|everywhere
 * align=right|everywhere


 * rowspan=3 bgcolor=#d0f0d0 align=center| f:X→Y
 * function arrow
 * rowspan=3|f: X → Y means the function f maps the set X into the set Y.
 * rowspan=3|Let f: ℤ → ℕ be defined by f(x) := x2.
 * align=center|from … to
 * align=right|set theory
 * align=center|from … to
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| o
 * function composition
 * rowspan=3|f o g is the function, such that (f o g)(x) = f(g(x)).
 * rowspan=3|if f(x) := 2x, and g(x) := x + 3, then (f o g)(x) = 2(x + 3).
 * align=center|composed with
 * align=right|set theory
 * align=center|composed with
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ℕ N
 * natural numbers
 * rowspan=3|N means { 1, 2, 3, ...}, but see the article on natural numbers for a different convention.
 * rowspan=3|ℕ = {| a| : a ∈ ℤ, a ≠ 0}
 * align=center|N
 * align=right|numbers
 * align=center|N
 * align=right|numbers
 * align=right|numbers


 * rowspan=3 bgcolor=#d0f0d0 align=center| ℤ Z ||integers
 * rowspan=3|ℤ means {..., &minus;3, &minus;2, &minus;1, 0, 1, 2, 3, ...} and ℤ+ means {1, 2, 3, ...} = ℕ.
 * rowspan=3|ℤ = {p, -p : p ∈ ℕ} ∪ {0}
 * align=center|Z
 * align=right|numbers
 * align=center|Z
 * align=right|numbers
 * align=right|numbers


 * rowspan=3 bgcolor=#d0f0d0 align=center| ℚ Q ||rational numbers
 * rowspan=3|ℚ means {p/q : p ∈ ℤ, q ∈ ℕ}.
 * rowspan=3|3.14000... ∈ ℚ π ∉ ℚ
 * align=center|Q
 * align=right|numbers
 * align=center|Q
 * align=right|numbers
 * align=right|numbers


 * rowspan=3 bgcolor=#d0f0d0 align=center| ℝ R ||real numbers
 * rowspan=3|ℝ means the set of real numbers.
 * rowspan=3|ℝ means the set of real numbers.


 * rowspan=3|π ∈ ℝ √(&minus;1) ∉ ℝ
 * align=center|R
 * align=right|numbers
 * align=right|numbers
 * align=right|numbers


 * rowspan=6 bgcolor=#d0f0d0 align=center| ℂ C ||complex numbers
 * rowspan=3|ℂ means {a + b i : a,b ∈ ℝ}.
 * rowspan=3|i = √(&minus;1) ∈ ℂ
 * align=center|C
 * align=right|numbers
 * arbitrary constant
 * rowspan=3| C can be any number, most likely unknown; usually occurs when calculating antiderivatives.
 * rowspan=3|if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C
 * align=center|C
 * align=right|integral calculus
 * rowspan=3| C can be any number, most likely unknown; usually occurs when calculating antiderivatives.
 * rowspan=3|if f(x) = 6x² + 4x, then F(x) = 2x³ + 2x² + C
 * align=center|C
 * align=right|integral calculus
 * align=right|integral calculus
 * align=right|integral calculus


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∞
 * infinity
 * rowspan=3|∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits.
 * rowspan=3|limx→0 1/|x| = ∞
 * align=center|infinity
 * align=right|numbers
 * align=center|infinity
 * align=right|numbers
 * align=right|numbers


 * rowspan=3 bgcolor=#d0f0d0 align=center| π ||pi
 * rowspan=3|π is the ratio of a circle's circumference to its diameter. Its value is 3.14159265....
 * rowspan=3|A = π r² is the area of a circle with radius r π radians = 180°  π ≈ 22 / 7
 * align=center|pi
 * align=right|Euclidean geometry
 * align=center|pi
 * align=right|Euclidean geometry
 * align=right|Euclidean geometry


 * rowspan=3 bgcolor=#d0f0d0 align=center| || … ||
 * norm
 * rowspan=3| || x || is the norm of the element x of a normed vector space.
 * rowspan=3| || x + y || ≤  || x ||  +   || y ||
 * align=center|norm of length of
 * align=right| linear algebra
 * align=center|norm of length of
 * align=right| linear algebra
 * align=right| linear algebra

$$\sum_{k=1}^{n}{a_k}$$ means a1 + a2 + … + an. $$\sum_{k=1}^{4}{k^2}$$ = 12 + 22 + 32 + 42
 * rowspan=3 bgcolor=#d0f0d0 align=center| ∑
 * summation
 * rowspan=3|
 * rowspan=3|
 * rowspan=3|
 * = 1 + 4 + 9 + 16 = 30


 * align=center|sum over … from … to … of
 * align=right|arithmetic
 * align=right|arithmetic
 * align=right|arithmetic

$$\prod_{k=1}^na_k$$ means a1a2···an. $$\prod_{k=1}^4(k+2)$$ = (1+2)(2+2)(3+2)(4+2)
 * rowspan=6 bgcolor=#d0f0d0 align=center| ∏
 * product
 * rowspan=3|
 * rowspan=3|
 * rowspan=3|
 * = 3 &times; 4 &times; 5 &times; 6 = 360


 * align=center|product over … from … to … of
 * align=right|arithmetic
 * align=right|arithmetic
 * align=right|arithmetic

$$\prod_{i=0}^{n}{Y_i}$$ means the set of all (n+1)-tuples
 * Cartesian product
 * rowspan=3|
 * rowspan=3|
 * (y0, …, yn).

$$\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3$$
 * rowspan=3|
 * align=center|the Cartesian product of; the direct product of
 * align=right|set theory
 * align=right|set theory
 * align=right|set theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∐
 * coproduct
 * rowspan=3|
 * rowspan=3|
 * align=center|coproduct over … from … to … of
 * align=right|category theory
 * align=center|coproduct over … from … to … of
 * align=right|category theory
 * align=right|category theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ′
 * derivative
 * rowspan=3|f ′(x) is the derivative of the function f at the point x, i.e., the slope of the tangent to f at x.
 * rowspan=3|If f(x) := x2, then f ′(x) = 2x
 * align=center|… prime derivative of
 * align=right|calculus
 * align=center|… prime derivative of
 * align=right|calculus
 * align=right|calculus


 * rowspan=6 bgcolor=#d0f0d0 align=center| ∫
 * indefinite integral or antiderivative
 * rowspan=3|∫ f(x) dx means a function whose derivative is f.
 * rowspan=3| ∫x2 dx = x3/3 + C
 * align=center|indefinite integral of the antiderivative of
 * align=right|calculus
 * align=center|indefinite integral of the antiderivative of
 * align=right|calculus
 * align=right|calculus


 * definite integral
 * rowspan=3|∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b.
 * rowspan=3|∫0b x2 dx = b3/3;
 * align=center|integral from … to … of … with respect to
 * align=right|calculus
 * align=center|integral from … to … of … with respect to
 * align=right|calculus
 * align=right|calculus


 * rowspan=3 bgcolor=#d0f0d0 align=center| ∇
 * gradient
 * rowspan=3|∇f (x1, …, xn) is the vector of partial derivatives (∂f / ∂x1, …, ∂f / ∂xn).
 * rowspan=3|If f (x,y,z) := 3xy + z², then ∇f = (3y, 3x, 2z)
 * align=center|del, nabla, gradient of
 * align=right|calculus
 * align=center|del, nabla, gradient of
 * align=right|calculus
 * align=right|calculus


 * rowspan=6 bgcolor=#d0f0d0 align=center| ∂
 * partial derivative
 * rowspan=3| With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant.
 * rowspan=3| If f(x,y) := x2y, then ∂f/∂x = 2xy
 * align=center|partial derivative of
 * align=right|calculus
 * align=center|partial derivative of
 * align=right|calculus
 * align=right|calculus


 * boundary
 * rowspan=3| ∂M means the boundary of M
 * rowspan=3| ∂{x : || x || ≤ 2} = {x : || x || = 2}
 * align=center|boundary of
 * align=right|topology
 * align=center|boundary of
 * align=right|topology
 * align=right|topology


 * rowspan=6 bgcolor=#d0f0d0 align=center| &perp;
 * perpendicular
 * rowspan=3|x &perp; y means x is perpendicular to y; or more generally x is orthogonal to y.
 * rowspan=3|If l &perp; m and m &perp; n then l || n.
 * align=center|is perpendicular to
 * align=right|geometry
 * align=center|is perpendicular to
 * align=right|geometry
 * align=right|geometry


 * bottom element
 * rowspan=3|x = &perp; means x is the smallest element.
 * rowspan=3|&forall;x : x ∧ &perp; = &perp;
 * align=center|the bottom element
 * align=right|lattice theory
 * align=center|the bottom element
 * align=right|lattice theory
 * align=right|lattice theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ||
 * parallel
 * rowspan=3|x || y means x is parallel to y.
 * rowspan=3|If l || m and m &perp; n then l &perp; n.
 * align=center|is parallel to
 * align=right|geometry
 * align=center|is parallel to
 * align=right|geometry
 * align=right|geometry


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⊧
 * entailment
 * rowspan=3| A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true.
 * rowspan=3| A ⊧ A ∨ ¬A
 * align=center|entails
 * align=right| model theory
 * align=center|entails
 * align=right| model theory
 * align=right| model theory


 * rowspan=3 bgcolor=#d0f0d0 align=center| ⊢
 * inference
 * rowspan=3|x ⊢ y means y is derived from x.
 * rowspan=3| A → B ⊢ ¬B → ¬A
 * align=center|infers or is derived from
 * align=right|propositional logic, predicate logic
 * align=center|infers or is derived from
 * align=right|propositional logic, predicate logic
 * align=right|propositional logic, predicate logic


 * rowspan=3 bgcolor=#d0f0d0 align=center| ◅
 * normal subgroup
 * rowspan=3| N ◅ G means that N is a normal subgroup of group G.
 * rowspan=3| Z(G) ◅ G
 * align=center|is a normal subgroup of
 * align=right|group theory
 * align=center|is a normal subgroup of
 * align=right|group theory
 * align=right|group theory


 * rowspan=6 bgcolor=#d0f0d0 align=center| /
 * quotient group
 * rowspan=3| G/H means the quotient of group G modulo its subgroup H.
 * rowspan=3| {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}}
 * align=center| mod
 * align=right| group theory
 * align=center| mod
 * align=right| group theory
 * align=right| group theory


 * quotient set
 * rowspan=3| A/~ means the set of all ~ equivalence classes in A.
 * align=center|
 * align=right| set theory
 * align=center|
 * align=right| set theory
 * align=right| set theory

where Q is the quaternion group and V is the Klein four-group.
 * rowspan=6 bgcolor=#d0f0d0 align=center| ≈
 * isomorphism
 * rowspan=3| G ≈ H means that group G is isomorphic to group H
 * rowspan=3| Q / {1, &minus;1} ≈ V,
 * rowspan=3| Q / {1, &minus;1} ≈ V,
 * align=center | is isomorphic to
 * align=right| group theory
 * approximately equal
 * rowspan=3|x ≈ y means x is approximately equal to y
 * rowspan=3|π ≈ 3.14159
 * align=center|is approximately equal to
 * align=right|everywhere
 * rowspan=3 bgcolor=#d0f0d0 align=center| ~
 * same order of magnitude
 * rowspan=3| m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use  ≈ .)
 * rowspan=3|2 ~ 5 8 × 9 ~ 100  but π2 ≈ 10
 * align=right|roughly similar poorly approximates
 * align=right|Approximation theory
 * rowspan=3 bgcolor=#d0f0d0 align=center| ~
 * same order of magnitude
 * rowspan=3| m ~ n, means the quantities m and n have the general size. (Note that ~ is used for an approximation that is poor, otherwise use  ≈ .)
 * rowspan=3|2 ~ 5 8 × 9 ~ 100  but π2 ≈ 10
 * align=right|roughly similar poorly approximates
 * align=right|Approximation theory
 * align=right|Approximation theory
 * align=right|Approximation theory

 = 2×-1 + 3×5 = 13
 * rowspan=3 bgcolor=#d0f0d0 align=center| <,>
 * inner product
 * rowspan=3| means the inner product between x and y, as defined in an inner product space.
 * rowspan=3|The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
 * rowspan=3|The standard inner product between two vectors x = (2, 3) and y = (-1, 5) is:
 * align=center|inner product of
 * align=right|vector algebra
 * align=right|vector algebra
 * align=right|vector algebra

{{1, 2, 3, 4}, {1, 2, 3, 4}, {2, 4, 6, 8}}
 * rowspan=3 bgcolor=#d0f0d0 align=center| ⊗
 * tensor product
 * rowspan=3| V ⊗ U means the tensor product of V and U.
 * rowspan=3| {1, 2, 3, 4} ⊗ {1,1,2} =
 * rowspan=3| {1, 2, 3, 4} ⊗ {1,1,2} =
 * align=center| tensor product of
 * align=right| linear algebra
 * }
 * align=right| linear algebra
 * }